Recently, an AlGaN/GaN.FET being a compound semiconductor device utilizing a hetero junction of the AlGaN/GaN and using GaN as the electron transit layer is actively developed (see, for example, Patent Documents 1 to 4). GaN is a material having a wide bandgap, a high breakdown field strength and a high saturation electron speed and is therefore very hopeful as a material of the semiconductor device achieving high-voltage operation and a high output. In the AlGaN/GaN.FET, the source electrode, the drain electrode, and the gate electrode are formed on the electron supply layer made of AlGaN and a protective film (for example, a SiN film) for reducing the surface trap is formed on the entire surface.    Patent Document 1: Japanese Laid-open Patent Publication No. 2007-227884    Patent Document 2: Japanese Laid-open Patent Publication No. 2004-288952    Patent Document 3: Japanese Laid-open Patent Publication No. 8-162476    Patent Document 4: Japanese Laid-open Patent Publication No. 4-282841
However, in the above-described AlGaN/GaN.FET, gate leakage larger than expected often occurs to make it difficult for the AlGaN/GaN.FET to perform high-voltage operation that is considered as an advantage thereof. Therefore, it is necessary to devise reduction of the gate leakage current in the AlGaN/GaN.FET, but any effective efficient measure has not been taken at present.